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Number Theory pp 179-192 | Cite as

The Elementary Proof of the Prime Number Theorem: An Historical Perspective

  • D. Goldfeld

Abstract

The study of the distribution of prime numbers has fascinated mathematicians since antiquity. It is only in modern times, however, that a precise asymptotic law for the number of primes in arbitrarily long intervals has been obtained. For a real number x > 1, let π(x) denote the number of primes less than x. The prime number theorem is the assertion that
$$ \mathop {\lim }\limits_{x \to \infty } \pi \left( x \right){\raise0.7ex\hbox{${}$} \!\mathord{\left/ {\vphantom {{} {}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${}$}}\frac{x} {{\log (x)}} = 1. $$
This theorem was conjectured independently by Legendre and Gauss.

Keywords

Prime Number Elementary Proof Tauberian Theorem Joint Paper Prime Number Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media New York 2004

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  • D. Goldfeld

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